# Triple solutions for nonlinear singular m-point boundary value problem

Volume 4, Issue 4, pp 262--269
• 866 Views

### Authors

Fuli Wang - School of Mathematics and Physics, Changzhou University, Changzhou, China.

### Abstract

In this paper, we study the existence of three solutions to the following nonlinear m-point boundary value problem $\begin{cases} u''(t) + \beta^2u(t) = h(t)f(t, u(t)),\,\,\,\,\, 0 < t < 1,\\ u'(0) = 0, u(1) =\Sigma^{m-2}_{i=1}\alpha_i u(\eta_i), \end{cases}$ where $0<\beta<\frac{\pi}{2}, f\in C([0,1]\times \mathbb{R}^+, \mathbb{R}^+). h(t)$ is allowed to be singular at $t = 0$ and $t = 1$. The arguments are based only upon the Leggett-Williams fixed point theorem. We also prove nonexist results.

### Keywords

• m-point boundary value problem
• Positive solutions
• Fixed point theorem.

•  34B15
•  34B25

### References

• [1] A. V. Bitsadze, On the theory of nonlocal boundary value problems, Soviet Math. Dokl. , 30 (1984), 8–10.

• [2] A. V. Bitsadze, A. A. Samarskii, On a class of conditionally solvable nonlocal boundary value problems for harmonic functions, Soviet Math. Dokl. , 31 (1985), 91–94.

• [3] V. A. Il’in, E. I. Moiseev, Nonlocal boundary value problems of the second kind for a Sturm- CLiouville operator, J. Differential Equations , 23 (1987), 979–987.

• [4] C. P. Gupta , Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl. , 168 (1992), 540–551.

• [5] F. Zhang, Multiple positive solutions for nonlinear singular third order boundary value problem in abstract spaces, J. Nonlinear. Sci. Appl., 1(1) (2008), 36–44.

• [6] X. R. Wu, F. Wang, Nonlinear solutions of singular second order three-point boundary value problem at resonance, J. Nonlinear. Sci. Appl., 1(1) (2008), 49–55.

• [7] Y. Y. Pang, Z. B. Bai, Existence and multiplicity of positive solutions for p-Laplacian boundary value problem on time scales, J. Nonlinear. Sci. Appl. , 3(1) (2010), 32–38.

• [8] A. Guezane-Lakoud, S. Kelaiaia, Solvability of a nonlinear boundary value problem, J. Nonlinear. Sci. Appl. , 4(4) (2011), 247–261.

• [9] F. Wang, Y. Cui, F. Zhang, Existence of nonnegative solutions for second order m-point boundary value problems at resonance, Appl. Math. Comput. , 217 (2011), 4849–4855.

• [10] G. Zhang, J. Sun, Positive solutions of m-point boundary value problems, J. Math. Anal. Appl. , 291 (2004), 406–418.

• [11] G. Zhang, J. Sun, Multiple positive solutions of singular second-orderm-point boundary value problems, J. Math. Anal. Appl. , 317 (2006), 442–447.

• [12] Y. Cui, Y. Zou, Nontrivial solutions of singular superlinear m-point boundary value problems, Appl. Math.Comp. , 187 (2007), 1256–1264.

• [13] X. Han, Positive solutions for a three-point boundary value problem, Nonlinear Anal., 66 (3) (2007), 679–688.

• [14] L. X. Truong, L. T. P. Ngoc, N. T. Long, Positive solutions for an m-point boundary value problem, Electron. J. Differential Eqns., 111 (2008), 1–11.

• [15] R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Math. J., 28 (1979), 673–688.